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Properties

Label 2-6-3.2-c30-0-4
Degree $2$
Conductor $6$
Sign $-0.129 - 0.991i$
Analytic cond. $34.2085$
Root an. cond. $5.84880$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.31e4i·2-s + (1.42e7 − 1.86e6i)3-s − 5.36e8·4-s + 3.80e10i·5-s + (4.31e10 + 3.29e11i)6-s + 7.78e12·7-s − 1.24e13i·8-s + (1.98e14 − 5.29e13i)9-s − 8.80e14·10-s + 2.27e15i·11-s + (−7.63e15 + 9.98e14i)12-s + 1.13e16·13-s + 1.80e17i·14-s + (7.07e16 + 5.40e17i)15-s + 2.88e17·16-s − 4.86e18i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.991 − 0.129i)3-s − 0.500·4-s + 1.24i·5-s + (0.0916 + 0.701i)6-s + 1.63·7-s − 0.353i·8-s + (0.966 − 0.257i)9-s − 0.880·10-s + 0.544i·11-s + (−0.495 + 0.0648i)12-s + 0.221·13-s + 1.15i·14-s + (0.161 + 1.23i)15-s + 0.250·16-s − 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(34.2085\)
Root analytic conductor: \(5.84880\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :15),\ -0.129 - 0.991i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(3.441305298\)
\(L(\frac12)\) \(\approx\) \(3.441305298\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.31e4iT \)
3 \( 1 + (-1.42e7 + 1.86e6i)T \)
good5 \( 1 - 3.80e10iT - 9.31e20T^{2} \)
7 \( 1 - 7.78e12T + 2.25e25T^{2} \)
11 \( 1 - 2.27e15iT - 1.74e31T^{2} \)
13 \( 1 - 1.13e16T + 2.61e33T^{2} \)
17 \( 1 + 4.86e18iT - 8.19e36T^{2} \)
19 \( 1 - 1.09e19T + 2.30e38T^{2} \)
23 \( 1 - 4.33e20iT - 7.10e40T^{2} \)
29 \( 1 - 1.37e21iT - 7.44e43T^{2} \)
31 \( 1 + 7.68e21T + 5.50e44T^{2} \)
37 \( 1 + 8.83e22T + 1.11e47T^{2} \)
41 \( 1 - 1.91e24iT - 2.41e48T^{2} \)
43 \( 1 + 5.27e24T + 1.00e49T^{2} \)
47 \( 1 - 1.59e25iT - 1.45e50T^{2} \)
53 \( 1 + 6.96e24iT - 5.34e51T^{2} \)
59 \( 1 + 5.40e26iT - 1.33e53T^{2} \)
61 \( 1 + 9.06e26T + 3.62e53T^{2} \)
67 \( 1 - 2.89e27T + 6.05e54T^{2} \)
71 \( 1 - 2.12e27iT - 3.44e55T^{2} \)
73 \( 1 - 5.48e27T + 7.93e55T^{2} \)
79 \( 1 - 1.73e27T + 8.48e56T^{2} \)
83 \( 1 + 4.74e28iT - 3.73e57T^{2} \)
89 \( 1 + 2.15e29iT - 3.03e58T^{2} \)
97 \( 1 - 2.31e29T + 4.01e59T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.53819942197584426912205504407, −14.53591684416247928188853065859, −13.79430934225684632258039388643, −11.40609853342796078276394810718, −9.579521799460917413031397597683, −7.87719545490080525628692798637, −7.09004190637974849541760104711, −4.89457512920275360537482411659, −3.14399946982935412902459771211, −1.57976294701497699804968742945, 1.05881147999772516208583087233, 1.97181369551006165200223903492, 3.92094097689586676541238045436, 5.04696872829247324555747675928, 8.204282942980551590401220248293, 8.730111263338960612345855144024, 10.61001913270523674360296273301, 12.30341172482673651877432675673, 13.64916811371159778723500405584, 14.91174898254389383750861929558

Graph of the $Z$-function along the critical line